TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION ONDREJˇ ZINDULKA Abstract. For a mapping f : X → Y between metric spaces the function lipf : X → [0, ∞] defined by diam f(B(x,r)) lipf(x) = liminf r→0 r is termed the lower scaled oscillation or little lip function. We prove that, given any positive integer d and a locally compact set Ω ⊆ Rd with a nonempty interior, for a typical continuous function f : Ω → R the set {x ∈ Ω : lipf(x) > 0} has both Hausdorff and lower packing dimensions exactly d − 1, while the set {x ∈ Ω : lipf(x) = ∞} has non-σ-finite (d−1)- dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Cs¨ornyei [3], Hanson [11] and Buczolich, Hanson, Rmoutil and Z¨urcher [5]. It follows, e.g., that a graph of a typical function f ∈ C(Ω) is microscopic, and for a typical function f : [0, 1] → [0, 1] there are sets A, B ⊆ [0, 1] of lower packing and Hausdorff dimension zero, respectively, such that the graph of f is contained in the set A × [0, 1] ∪ [0, 1] × B. 1. Introduction For a mapping f : X Y between metric spaces the function lipf : X [0, ] defined by → → ∞ diam f(B(x, r)) (1) lipf(x) = lim inf r→0 r is termed the lower scaled oscillation or little lip function. (B(x, r) denotes the closed ball of radius r centered at x.) Let us mention that some authors, cf. [3], define the scaled oscillations from the version of ωf (cf.(6)) given by ωf (x, r) = sup f(y) f(x) that may be more suitable for, e.g., differentiation con- y∈B(x,r)| − | siderations. It is clear though that ωf (x, r) 6 ωf (x, r) 6 2ω(x, r) andb thus the two lower scaled oscillation functions differ at most by a factor of 2. Since we are arXiv:1910.14527v1 [math.CA] 31 Oct 2019 interested only in the sets lipf = 0b, lipf > 0 and lipfb = , it does not matter which of the two definitions{ we} use.{ } { ∞} We are interested in the behavior of lipf for continuous functions on a locally compact set in the d-dimensional Euclidean space. There is also the upper scaled oscillation function (or the big lip function) diam f(B(x, r)) Lipf(x) = lim sup r→0 r 2000 Mathematics Subject Classification. 26B05, 26B35, 26A27, 28A78, 26A16. Key words and phrases. lower scaled oscillation, continuous function, typical function, Haus- dorff measure, Hausdorff dimension, lower packing dimension. The author was supported from European Regional Development Fund-Project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16 019/0000778). 1 TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION 2 that plays a crucial role in the famous Stepanov-Rademacher Theorem that asserts that if f is measurable, then it is differentiable at almost every point where Lipf is finite. The lower scaled oscillation, lipf, was used in [6] and [17] in the same context: in the study of validity of the Rademacher Theorem in metric spaces. Recently there has been a lot of interest in the behavior of functions at points where lipf(x) is finite and in particular in functions that have only a few points with lipf(x)= . In particular, the differentiability of such functions has been studied and the structure∞ of exceptional sets x : lipf(x) = has been investigated. Balogh and Cs¨ornyei brought deep results{ on lipf in∞} [3] and ignited thus more interest in the notion. The subject was further studied in [12, 13, 18, 5, 11]. The papers [3, 5, 11] are of particular importance for the present paper. In [3], Balogh and Cs¨ornyei constructed an example contrasting the Rademacher-Stepanov Theorem: Theorem ([3, Theorem 1.4]). There is a continuous function f : [0, 1] R that has lipf =0 almost everywhere and yet is nowhere differentiable. → This result was improved in two directions. Hanson’s [11] example has even a smaller exceptional set: Theorem ([11, Theorem 2.3]). There is a continuous function f : [0, 1] R that has lipf =0 everywhere except on a set of Hausdorff dimension zero and→ yet it is nowhere differentiable. In another direction, Buczolich, Hanson, Rmoutil and Z¨urcher proved in [5] that the example of Balogh and Csornyei is actually typical (which sharply contrasts the classical result of Banach [4] that a typical continuous function has Lipf = at each point). Recall that a property of continuous functions is called typical (or∞ generic) if it holds for all functions in a dense Gδ-set in C([0, 1]). Theorem ([5, Theorem 4.2]). A typical continuous function f : [0, 1] R has lipf =0 almost everywhere. → Thus a typical function has lipf = 0 a.e. and is nowhere differentiable. Both of the latter results have also higher dimensional versions. The aim of this paper is to prove a sharp theorem roofing the above three results. In its simplest form it reads as follows. (The full strength version is presented in Section 3). Theorem 1.1. For a typical continuous function f : Ω R, where Ω Rd is a locally compact set, the exceptional sets → ⊆ x Ω : lipf(x) > 0 , x Ω : lipf(x)= { ∈ } { ∈ ∞} have lower packing dimension (and thus also Hausdorff dimension) at most d 1. If, moreover, Ω has nonempty interior, then the two exceptional sets have− both Hausdorff and lower packing dimension exactly d 1, and are both of non-σ-finite (d 1)-dimensional Hausdorff measure. − − This theorem and some consequences are proved in Section 3. The above theorems yields a perhaps surprising typical property that asserts that typical functions admit a very small set whose image is very large. Theorem 1.2. For a typical continuous function f : Ω R, where Ω Rd is a locally compact set, there is a partition A B =Ω such that→ ⊆ ∪ TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION 3 (i) the lower packing dimension of A is at most d 1, (ii) the Hausdorff dimension of f(B) is 0. − The following restatement of the latter is maybe even more interesting: a graph of a typical function on [0, 1] is contained in a rather sparse grid of horizontal and vertical lines. Corollary 1.3. For a typical function f C([0, 1]) there are sets A, B [0, 1] ∈ ⊆ such that dimP A =0, dimH B =0 and such that Graph(f) (A R) (R B). ⊆ × ∪ × These facts regarding the size of a typical graph are discussed in Section 4. In Section 5 we prove that the exceptional sets and the graph of a typical function are microscopic, a notion that has been recently studied in a number of papers, see Definition 5.1. 2. Fractal measures and dimensions In this section we review some background and preliminary material. Namely we recall the notions Hausdorff and lower packing dimensions and corresponding measures. Some of the common notation includes B(x, r) for the closed ball centered at x, with radius r; d is a generic symbol for a metric; we write diam E for the diameter of a set E in a metric space. Letters d,n,m,i,j,k are generic symbols for positive integers. R denotes the real line and Rd the d-dimensional Euclidean space; ω stands for the set of natural numbers including zero and N+ stands for the set of natural numbers excluding zero. A denotes the cardinality (finite or infinite) of the set A. We write X X for an| increasing| sequence of sets X with union X. nր h ni Hausdorff measure. Besides standard Hausdorff measures we will also make use of those induced by gauges. A non-decreasing, right-continuous function ψ : [0, ) [0, ) such that ψ(0) = 0 and ψ(r) > 0 if r> 0 is called a gauge. The following∞ is→ the∞ common (partial) ordering of gauges, cf. [22]: def ψ(r) φ 4 ψ lim sup < . ≡ r→0+ φ(r) ∞ In the case when ψ(r) = rs for some s > 0, we write φ 4 s instead of φ 4 ψ and likewise for φ < s. If δ > 0, a cover of a set E X is termed a δ-cover if diam A 6 δ for all A . If ψ is a gauge,A the ψ-dimensional⊆ Hausdorff measure H ψ(E) of a set E ∈X A is defined thus: For each δ > 0 set ⊆ ψ H (E) = inf ψ(diam En): En is a countable δ-cover of E δ n { } n o and put X H ψ H ψ (E) = sup δ (E). δ>0 In the common case when ψ(r)= rs for some s> 0, we write H s for H ψ, and the same licence is used for other measures and set functions arising from gauges. We also define the singular case of H 0 to be the counting measure. Properties of Hausdorff measures are well-known, see e.g. in [22]. TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION 4 Hausdorff dimension. Recall that the Hausdorff dimension of E is denoted and defined by s dimH E = sup s : H (E) > 0 . { } Properties of Hausdorff dimension are well-known, see, e.g., [9]. Lower box-counting measure. Besides Hausdorff dimension we will also make use of the lower packing dimension. There are several ways to define it. We do so via the so called lower box-counting measures. For E X and δ > 0, define ⊆ (2) N (E) = inf : is δ-cover of E . δ {|E| E } The mapping δ Nδ(E) is sometimes referred to as the box-counting function of E.